Problem: Let $S$ be a surface in 3D described by the equation $z = ye^{2x} - y^2$. What is the equation of the plane tangent to $S$ at $(1, 5)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $z = 5e^2 - 25 + 10e^2(x - 1) + (e^2 - 10)(y - 5)$ (Choice B) B $z = 25 - 5e^2 + 10e^2(x - 1) + (e^2 - 10)(y - 5)$ (Choice C) C $z = 5e^2 - 25 + 5e^2(x - 1) + e^2(y - 5)$ (Choice D) D $z = 10e^2(x + 1) + (e^2 - 10)(y + 5)$
Solution: The equation for a tangent plane of an explicitly defined surface $z = f(x, y)$ at the point $(a, b)$ is: $f(a, b) + f_x(x - a) + f_y(y - b) = z$ [What's the intuition behind the formula?] Let's find $f(1, 5)$, $f_x$, and $f_y$. $\begin{aligned} &f(1, 5) = 5e^2 - 25 \\ \\ &f_x = 2ye^{2x} = 10e^2 \\ \\ &f_y = e^{2x} - 2y = e^2 - 10 \end{aligned}$ Putting it all together, here's the equation for the tangent plane of $S$ at $(1, 5)$ : $z = 5e^2 - 25 + 10e^2(x - 1) + (e^2 - 10)(y - 5)$